Residual operators of uninorms

Abstract

 Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class U _min) and maximum operator (the class U _max) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class U _min and of the residual coimplicator of members of the class U _max is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.